# Cancellation Laws in a Ring

Cancellation Laws in a Ring

We say that cancellation laws hold in a ring $R$ if
$ab = bc\,\left( {a \ne 0} \right)\,\, \Rightarrow \,b = c$ and $ba = ca\,\,\left( {a \ne 0} \right)\,\,\, \Rightarrow b = c$ where $a,b,c$ are in $R$.

Thus in a ring with zero divisors, it is impossible to define a cancellation law.

Theorem:

A ring has no divisor of zero if and only if the cancellation laws holds in $R$.

Proof:

Suppose that $R$ has no zero divisors. Let $a,b,c$ be any three elements of $R$ such that $a \ne 0,\,\,ab = ac$.

Now
$\begin{gathered} ab = ac\, \Rightarrow ab – ac = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a\left( {b – c} \right) = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b – c = 0\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{because}}\,R\,{\text{is}}\,{\text{without}}\,{\text{zero}}\,{\text{divisor}}\,{\text{and}}\,a \ne 0} \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b = c \\ \end{gathered}$

Thus the left cancellation law holds in $R$. Similarly, it can be shown that the right cancellation law also holds in $R$.

Conversely, suppose that the cancellation law holds in $R$. Let $a,b \in R$ and if possible let $ab = 0$ with $a \ne 0,\,b \ne 0$ then $ab = a \cdot 0$ (because $a \cdot 0 = 0$).

Since $a \ne 0,\,\,ab = a \cdot 0 \Rightarrow b = 0$

Hence we get a contradiction to our assumption that $b \ne 0$ and therefore the theorem is established.

Division Ring

A ring is called a division ring if its non-zero elements form a group under the operation of multiplication.

Pseudo Ring

A non-empty set $R$ with binary operations $+$ and $\times$ satisfying all the postulates of a ring except right and left distribution laws is called pseudo ring if
$\left( {a + b} \right) \cdot \left( {c + d} \right) = a \cdot c + a \cdot d + b \cdot c + b \cdot d$ for all $a,b,c,d \in R$